This section shows that a database instance \(\mathcal{J}\xrightarrow{I}\mathbf{Set}\) is a diagram in Set. Also the functor \(\Pi_!\) takes the limit of this diagram.
Details not presented, but it suffices to work with just the finitely-presented graph of \(\mathcal{J}\) rather than the category itself (path equations not relevant for this).
Let \(\mathcal{J}\) be a category presented by the finite graph \((\{v_1,...,v_n\},A,s,t)\) with some equations.
Let \(\mathcal{J}\xrightarrow{D}\mathbf{Set}\) be some set-valued functor.
The set \(\underset{\mathcal{J}}{lim}D := \{(d_1,...,d_n)\ |\ \forall i: d_i \in D(v_i)\ \text{and}\ \forall v_i\xrightarrow{a}v_j \in A: D(a)(d_i)=d_j\}\)
... together with projection maps \(lim_\mathcal{J}D \xrightarrow{p_i}D(v_i)\) given by \(p_i(d_1,...,d_n):=d_i\)
... is a limit of \(D\). NOCARD
NOT PROVEN
With respect to Proposition 3.95, if \(\mathcal{J}=0\), then \(n=0\); there are no vertices.
There is exactly one empty tuple which vacuously satisfies the properties, so we’ve constructed the limit as the singleton set \(\{()\}\) consisting of just the empty tuple.
Thus the limit of the empty diagram, i.e. the terminal object in Set, is the singleton set.
If \(\mathcal{J}\) is presented by the cospan graph \(\boxed{\overset{x}\bullet \xrightarrow{f} \overset{a}\bullet \xleftarrow{g}\overset{y}\bullet}\) then its limit is known as a pullback.
Given the diagram \(X \xrightarrow{f}A\xleftarrow{g}Y\), the pullback is the cone shown below:
Because the diagram commutes, the diagonal arrow is superfluous. One can denote pullbacks instead like so:
The pullback picks out the \((X,Y)\) pairs which map to the same output.
Show that the limit formula works for products in Set
The diagram, whose limit is a product, is \(\mathcal{J}=\boxed{\overset{v}\bullet\ \overset{w}\bullet}\) (see Exaample 3.94)
\(V=\{v,w\}, A=\varnothing\)
The second condition of the set comprehension is vacuously satisfied because \(A = \varnothing\)
So all we have left is all pairs of tuples where the first element comes from \(D(v)\) and the second element comes from the set \(D(w)\).
This is the Cartesian product \(D(v) \times D(w)\)
If \(1 \xrightarrow{D}\mathbf{Set}\) is a functor, what is the limit of \(D\)? Compute using the limit formula and check answer against the limit definition.